Cut chicken into uniform serving size pieces (2-3 per breast) and salt and pepper each piece liberally and allow to sit for 5 minutes.
Heat the vegetable oil and butter in heavy large skillet over medium heat.
In a medium bowl, mix the salt and pepper in the all purpose flour and then dip the chicken in the flour - pat the flour into the chicken breast pieces and then sear the chicken in the hot oil until each side is golden colored. Remove from the pan and set on a plate until all of the chicken is browned on both sides.
Add the extra vegetable oil to hot skillet. Saute mushrooms and garlic until mushrooms begin to brown.
Add white wine to skillet all at once, it will probably make a noise and some steam. Make sure to scrap up any brown bits on the pan - these add FLAVOR! Add thyme to mushroom/wine mixture in pan.
Add the chicken back to the pan. Bring to a boil, then reduce heat, cover and simmer for 10-15 minutes until the chicken is cooked through and juices run clear.
Remove chicken from pan. Add the cream and heat through. Add the parmesan cheese.
Cook, stirring constantly over low heat until cheese melts. Continue cooking until sauce is reduced by about ½.
Garnish with sprigs of fresh thyme.